Price, who recorded his third shutout in a 3-0 win over Winnipeg on Sunday, denied Jason Williams on Pittsburgh's final shootout attempt after making 32 saves during the game.
"Carey was great again tonight," said Plekanec, who snapped a shot in off the post. "He gave us a chance to stay in the game and he was great again in the shootout. We were able to pay him back with that performance in the shootout."
Montreal won for just the second time in nine shootouts this season.
"We blew two leads but at the end of the day I thought we kept a really positive attitude on the bench," Price said.
Evgeni Malkin, who extended his NHL lead to 62 points with an assist on James Neal's goal in the third, used a spin-around move to convert the Penguins' second attempt.
"I think I almost got a piece of it," Price said. "But with that big, long reach and those hands, he's got a lot of cats in the bag."
The Penguins fell to 7-3 in shootouts.
"It was good for the crowd but I was disappointed to lose that one," said Fleury, who turned aside five of eight shootout attempts after making 28 saves in regulation.
Louis Leblanc scored in the second period to put the Canadiens up 1-0. Eller restored Montreal's lead at 2-1 with his 11th goal 3 minutes into the third.
Neal scored his 28th goal 7:32 into the third as Pittsburgh erased a pair of one-goal deficits in the period.
Pascal Dupuis drew Pittsburgh even at 1 early in the third when he drove the net to score a short-handed goal.
Pittsburgh coach Dan Bylsma called his timeout with a faceoff in the Canadiens' zone and 18.3 seconds left in overtime.
Malkin got a minor for roughing and Canadiens forward Mathieu Darche was called for interference.
"I was just stepping in for the best player in the world right now, and the best player in the NHL, so just making sure that we send a message that they're not going to touch our players," said Letang, who was not penalized.
Price made a pad save on Kunitz's quick shot 7 minutes into the second on a setup off a faceoff won by Malkin in Montreal's zone.
The Canadiens' goalie lost his bid for a second straight shutout 1:03 into the third when Dupuis was credited with a short-handed goal that tied it at 1. Joe Vitale's centering pass from the right side went in off Dupuis' right skate as he drove to the net. A brief video review upheld the call.
Fleury came up with a few big stops of his own, including against Leblanc on a partial breakaway 13 minutes in. He also turned aside Plekanec's shot on a short-handed 3-on-1 early in the second.
Plekanec was denied by Fleury again when he came out of the penalty box to take a breakaway pass right after Subban stopped Malkin's shot at an otherwise open net. Price was caught out of position when he fell after coming out to play the puck late in the Penguins' power play.
Luck was against Fleury as the Canadiens regained the lead shortly after Kunitz's tying goal.
Eller took a shot from the left side just over the blue line that struck the right post. The puck bounded back off Fleury's backside and across the goal line to give Montreal its second lead.
"He had four or five really big saves, really good scoring chances by their team," Bylsma said of Fleury. "They had some short-handed opportunities, a breakaway coming out of the box there after the power play -- he made a number of huge saves for us and really stood strong. It was an unfortunate goal for the second one, but he played a very good game, making some big saves, especially in the first half of the game to keep it at zero."
Leblanc scored his third goal 11:21 into the second. It was his first in four games since he was recalled from AHL Hamilton.
- Williams was recalled from AHL Scranton/Wilkes-Barre earlier in the day. He had one goal and one assist in seven games with Pittsburgh earlier this season.
- Canadiens LW Travis Moen returned after missing five games because of an upper-body injury.
- Montreal D Yannick Weber went to the dressing room after Kunitz's check sent him into the boards 3:48 into overtime.